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Theorem isstruct2 16472
Description: The property of being a structure with components in (1st𝑋)...(2nd𝑋). (Contributed by Mario Carneiro, 29-Aug-2015.)
Assertion
Ref Expression
isstruct2 (𝐹 Struct 𝑋 ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)))

Proof of Theorem isstruct2
Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brstruct 16471 . . 3 Rel Struct
21brrelex12i 5580 . 2 (𝐹 Struct 𝑋 → (𝐹 ∈ V ∧ 𝑋 ∈ V))
3 ssun1 4124 . . . . 5 𝐹 ⊆ (𝐹 ∪ {∅})
4 undif1 4397 . . . . 5 ((𝐹 ∖ {∅}) ∪ {∅}) = (𝐹 ∪ {∅})
53, 4sseqtrri 3980 . . . 4 𝐹 ⊆ ((𝐹 ∖ {∅}) ∪ {∅})
6 simp2 1134 . . . . . . 7 ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → Fun (𝐹 ∖ {∅}))
76funfnd 6359 . . . . . 6 ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → (𝐹 ∖ {∅}) Fn dom (𝐹 ∖ {∅}))
8 elinel2 4148 . . . . . . . . . . 11 (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) → 𝑋 ∈ (ℕ × ℕ))
9 1st2nd2 7703 . . . . . . . . . . 11 (𝑋 ∈ (ℕ × ℕ) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
108, 9syl 17 . . . . . . . . . 10 (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
11103ad2ant1 1130 . . . . . . . . 9 ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
1211fveq2d 6647 . . . . . . . 8 ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → (...‘𝑋) = (...‘⟨(1st𝑋), (2nd𝑋)⟩))
13 df-ov 7133 . . . . . . . . 9 ((1st𝑋)...(2nd𝑋)) = (...‘⟨(1st𝑋), (2nd𝑋)⟩)
14 fzfi 13323 . . . . . . . . 9 ((1st𝑋)...(2nd𝑋)) ∈ Fin
1513, 14eqeltrri 2909 . . . . . . . 8 (...‘⟨(1st𝑋), (2nd𝑋)⟩) ∈ Fin
1612, 15eqeltrdi 2920 . . . . . . 7 ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → (...‘𝑋) ∈ Fin)
17 difss 4084 . . . . . . . . 9 (𝐹 ∖ {∅}) ⊆ 𝐹
18 dmss 5744 . . . . . . . . 9 ((𝐹 ∖ {∅}) ⊆ 𝐹 → dom (𝐹 ∖ {∅}) ⊆ dom 𝐹)
1917, 18ax-mp 5 . . . . . . . 8 dom (𝐹 ∖ {∅}) ⊆ dom 𝐹
20 simp3 1135 . . . . . . . 8 ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → dom 𝐹 ⊆ (...‘𝑋))
2119, 20sstrid 3954 . . . . . . 7 ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → dom (𝐹 ∖ {∅}) ⊆ (...‘𝑋))
2216, 21ssfid 8717 . . . . . 6 ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → dom (𝐹 ∖ {∅}) ∈ Fin)
23 fnfi 8772 . . . . . 6 (((𝐹 ∖ {∅}) Fn dom (𝐹 ∖ {∅}) ∧ dom (𝐹 ∖ {∅}) ∈ Fin) → (𝐹 ∖ {∅}) ∈ Fin)
247, 22, 23syl2anc 587 . . . . 5 ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → (𝐹 ∖ {∅}) ∈ Fin)
25 p0ex 5258 . . . . 5 {∅} ∈ V
26 unexg 7447 . . . . 5 (((𝐹 ∖ {∅}) ∈ Fin ∧ {∅} ∈ V) → ((𝐹 ∖ {∅}) ∪ {∅}) ∈ V)
2724, 25, 26sylancl 589 . . . 4 ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → ((𝐹 ∖ {∅}) ∪ {∅}) ∈ V)
28 ssexg 5200 . . . 4 ((𝐹 ⊆ ((𝐹 ∖ {∅}) ∪ {∅}) ∧ ((𝐹 ∖ {∅}) ∪ {∅}) ∈ V) → 𝐹 ∈ V)
295, 27, 28sylancr 590 . . 3 ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → 𝐹 ∈ V)
30 elex 3489 . . . 4 (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) → 𝑋 ∈ V)
31303ad2ant1 1130 . . 3 ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → 𝑋 ∈ V)
3229, 31jca 515 . 2 ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → (𝐹 ∈ V ∧ 𝑋 ∈ V))
33 simpr 488 . . . . 5 ((𝑓 = 𝐹𝑥 = 𝑋) → 𝑥 = 𝑋)
3433eleq1d 2896 . . . 4 ((𝑓 = 𝐹𝑥 = 𝑋) → (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ↔ 𝑋 ∈ ( ≤ ∩ (ℕ × ℕ))))
35 simpl 486 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → 𝑓 = 𝐹)
3635difeq1d 4074 . . . . 5 ((𝑓 = 𝐹𝑥 = 𝑋) → (𝑓 ∖ {∅}) = (𝐹 ∖ {∅}))
3736funeqd 6350 . . . 4 ((𝑓 = 𝐹𝑥 = 𝑋) → (Fun (𝑓 ∖ {∅}) ↔ Fun (𝐹 ∖ {∅})))
3835dmeqd 5747 . . . . 5 ((𝑓 = 𝐹𝑥 = 𝑋) → dom 𝑓 = dom 𝐹)
3933fveq2d 6647 . . . . 5 ((𝑓 = 𝐹𝑥 = 𝑋) → (...‘𝑥) = (...‘𝑋))
4038, 39sseq12d 3976 . . . 4 ((𝑓 = 𝐹𝑥 = 𝑋) → (dom 𝑓 ⊆ (...‘𝑥) ↔ dom 𝐹 ⊆ (...‘𝑋)))
4134, 37, 403anbi123d 1433 . . 3 ((𝑓 = 𝐹𝑥 = 𝑋) → ((𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥)) ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋))))
42 df-struct 16464 . . 3 Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
4341, 42brabga 5394 . 2 ((𝐹 ∈ V ∧ 𝑋 ∈ V) → (𝐹 Struct 𝑋 ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋))))
442, 32, 43pm5.21nii 383 1 (𝐹 Struct 𝑋 ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  Vcvv 3471  cdif 3907  cun 3908  cin 3909  wss 3910  c0 4266  {csn 4540  cop 4546   class class class wbr 5039   × cxp 5526  dom cdm 5528  Fun wfun 6322   Fn wfn 6323  cfv 6328  (class class class)co 7130  1st c1st 7662  2nd c2nd 7663  Fincfn 8484  cle 10653  cn 11615  ...cfz 12875   Struct cstr 16458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-1o 8077  df-oadd 8081  df-er 8264  df-en 8485  df-dom 8486  df-sdom 8487  df-fin 8488  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-nn 11616  df-n0 11876  df-z 11960  df-uz 12222  df-fz 12876  df-struct 16464
This theorem is referenced by:  structn0fun  16474  isstruct  16475  setsstruct2  16500
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